# Trace of a tensor to the power n, as a polynomial of it's invariants?

I am currently working through some exercises for a course, and a property of tensors was given. I was curious where it came from - so I tried to prove the property, but I can't seem to figure it out. Any help? The statement is as follows:

Suppose I have a second rank tensor $Q$, which has dimension three. I can define the following quantity: $$I_n = \mathrm{Tr}(Q^n)$$ Now we have three invariants; $I_1$, $I_2$, $I_3$. In general we can rewrite $I_n$ as a polynomial in $I_1$, $I_2$, and $I_3$.

I came as far as rewriting $I_n = \lambda_1^n + \lambda_2^n + \lambda_3^n$, but from there on I do not see how to continue. Simply filling this in to the polynomial did not really give me any leads.

• Cayley–Hamilton gives you a linear relationship between $I,Q^1,Q^2,Q^3$. – Chappers May 4 '18 at 20:10